On the symmetry of sequentiality

Abstract

We offer a symmetric account of sequentiality, by means of symmetric algorithms, which are pairs of sequential functions, mapping input data to output data, and output exploration trees to input exploration trees, respectively. We use the framework of sequential data structures, a reformulation of a class of Kahn-Plotkin's concrete data structures. In sequential data structures, data are constructed by alternating questions and answers. Sequential data structures and symmetric algorithms are the objects and morphisms of a symmetric monoidal closed category, which is also cartesian, and is such that the unit is terminal. Our category is a full subcategory of categories of games considered by Lamarche, and by Abramsky-Jagadeesan, respectively.

Following Lamarche, we construct a comonad corresponding to contraction. We define this comonad via an adjunction between the category of symmetric algorithms and the “old” cartesian closed category of sequential algorithms, defined in the late seventies by the author and Gérard Berry. Thus sequential algorithms model not only typed λ-calculus, but also intuitionistic affine logic, with connectives ⊗, 1, ⊸, x, and τ.

This work, while finding its roots in the study of sequentiality, presents striking correspondences with game-theoretic concepts, introduced by Blass in the early seventies in a very different context. The aim of the present work is to offer a systematic connection between sequentiality and games. Also, the notion of symmetric algorithm appears to be new.